Partitions

Description

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Model theory

Whiten-Beta

The Whiten-Beta expression for partition to overflow is:

$E_{\rm {oa}}({\bar {d}}_{i})=C\left[{\dfrac {\left(1+\beta \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)(\exp(\alpha )-1)}{\exp \left(\alpha \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $E_{\rm {oa}}({\bar {d}}_{i})$ is the fraction of particles of size interval $i$ in the feed reporting to the overflow stream (frac)
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)
• $d_{\rm {50c}}$ is the corrected size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
• $C$ is the fraction of feed liquids (or fines) split to overflow (frac)
• $\alpha$ is a parameter representing the sharpness of separation
• $\beta$ is a term introduced to accommodate the so-called fish-hook effect, and controls the initial rise in the efficiency curve at finer sizes
• $\beta ^{*}$ is computed to ensure the Whiten-Beta function preserves the definition of $d_{\rm {50c}}$ in the presence of the fish-hook, i.e. $E=0.5C$ at $d_{\rm {50c}}$

The value of $\beta ^{*}$ is not an input parameter and is computed numerically, by iteration, since by rearrangement:

$\beta ^{*}={\dfrac {\ln {\big [}\exp(\alpha )+2\beta \beta ^{*}(\exp(\alpha )-1){\big ]}}{\alpha }}$

When the value of $\beta$ is zero, the Whiten-Beta equation reverts to:

$E_{{\rm {oa}}i}=C\left[{\dfrac {\exp(\alpha )-1}{\exp \left(\alpha {\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]$

Excel

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SysCAD

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